The purpose of this chapter is to demonstrate the adequacy of many of the modal logics presented in this book. Remember, a system S is adequate when the arguments that can be proven in S and the S-valid arguments are exactly the same. When S is adequate, its rules pick out exactly the arguments that are valid according to its semantics, and so it has been correctly formulated. A proof of the adequacy of S typically breaks down into two parts, namely, to show (Soundness) and (Completeness).

1. (Soundness) If H ⊢S C then H ⊧S C.

2. (Completeness) If H ⊧S C then H ⊢S C.

Soundness of K

Let us begin by showing the soundness of K, the simplest propositional modal logic. We want to show that if an argument is provable in K (H ⊢K C), then it is K-valid (H ⊧K C). So assume that there is a proof in K of an argument H / C. Suppose for a moment that the proof involves only the rules of propositional logic (PL). The proof can be written in horizontal notation as a sequence of arguments, each of which is justified by (Hyp) or follows from previous entries in the sequence by one of the rules (Reit), (CP), (MP), or (DN). For example, here is a simple proof in PL of p→q / ∼∼p→q, along with the corresponding sequence of arguments written in horizontal form at the right.